Preview Activity 11.6.1.
Your self-driving car company, Steer Clear, is doing well and almost ready to launch its first car. Some of your engineers have reported that the car has problems when the air intake encounters too much large particulate matter in the air (like sand, dust, large pollen, smog, etc.). In order to fix this issue, you have created a new type of filter that uses a sophisticated mesh and gravity to filter out large particulate matter, but this new type of filter gets clogged if exposed to too much large particulate matter too quickly. So we will need to measure the rate at which your car’s intake will be exposed to large particulate matter per unit time.
You consult a friend Alex who does atmospheric modeling of large particulate matter in your area. Alex created \(P(x,y)=10-\frac{1}{2} x^2-\frac{1}{5}y^2\text{,}\) a function that describes the amount of large particular matter in the air in terms of location relative to Alex’s lab. In particular, the \(x\) coordinate is the distance East/West of from Alex’s lab in kilometers and the \(y\)-coordinate is the distance North/South from Alex’s lab in kilometers.
Since you have completed the self-driving part of your car, you know that your car will move along a test course as a function of time given by
\begin{equation*}
\vr(t) = \langle x(t), y(t) \rangle = \langle 2-t^2, t^3 + 1\rangle.
\end{equation*}
where the \(x\) and \(y\) coordinates are the same as measured by Alex’s function and \(t\) is measured in minutes.
(a)
Substitute \(x(t)\) and \(y(t)\) (from your self-driving car path) into your expression for \(P(x,y)\) to get an expression for \(P(t)\text{,}\) the amount of large particulate matter encountered at each location on your test course as a function of time driven. Do not simplify your expression for \(P(t)\text{.}\)
(b)
Use your derivative rules from single variable calculus to find \(P'(t)=\frac{dP}{dt}\text{.}\) Note that this is not a partial derivative because \(P(t)\) only has one dependent variable. You should NOT simplify your expression for \(P'(t)\text{.}\)
(c)
Alex has told you that the amount of large particulate matter in the air changes in the \(y\)-direction (North/South) because of the proximity to industrial pollution from a factory complex. Also, the amount of large particulate matter in the air changes in the \(x\)-direction (East/West) because of tree pollen from a particular kind of tree in a nearby forest. We would like to understand how the rate of large particulate matter encountered on our drive is split into these two factor (industrial pollution and tree pollen).
Alex suggests that you measure \(\frac{dP}{dt}\) at a location \((x,y)=(a,b)\) by taking the derivative of the linearization for \(P\) at \((a,b)\text{.}\) The linearization of \(P\) at \((a,b)\) is given by
\begin{equation*}
L(x,y)=P(a,b)+P_x(a,b)(x-a)+P_y(a,b)(y-b)
\end{equation*}
Remember that we will need to treat \(x\) and \(y\) as functions of time.
Write a few sentences to explain why
\begin{equation}
\frac{dP}{dt}\approx\frac{dL}{dt} = 0 + P_x(a,b) \frac{dx}{dt}+P_y(a,b) \frac{dy}{dt}\tag{11.6.1}
\end{equation}
and how each of \(P_x(a,b) \frac{dx}{dt}\) and \(P_y(a,b) \frac{dy}{dt}\) measure the rate of change in pollution coming from tree pollen and industrial pollution.
(d)
Compute each of the following and substitute your expressions into (11.6.1), Alex’s approximation. Do NOT simplify your expression, just substitute in the values for each element below.
-
\(\displaystyle P_x(a,b) \)
-
\(\displaystyle P_x(a,b) \)
-
\(\displaystyle \frac{dx}{dt} \)
-
\(\displaystyle \frac{dy}{dt} \)
(e)
Compare your results for part 11.6.1.b and part 11.6.1.d and write a few sentences that identifies how each part of your answer for part 11.6.1.d corresponds to different parts of part 11.6.1.b. Remember your work for part 11.6.1.b is completely in terms of \(t\) so you will need to translate some terms from \(t\) into its meaning in \(x\text{,}\) \(y\text{,}\) or others.
